Stress is a vector quantity, with the dimensions of force per unit area. It is useful to think of it in the same way as pressure (which is a special kind of stress that only acts in the normal direction to a surface).
Therefore, stress can only be defined relative to a given surface. Suppose we have a material medium on which we have applied some forces that caused it to deform. If you take an arbitrary point $P$ inside the medium and slice the medium at this point by some imaginary plane, then the stress relative to that plane is given by:
$$\vec t = \mathbf T \vec n,$$
where $\vec n$ is the normal unit vector to the plane and $\mathbf T$ is the Cauchy stress tensor. The stress vector $\vec t$ measures force per unit area acting on that plane at point $P$.
In general, the stress vector is not normal to the plane, and it can have a normal and a tangential component. Also, if you would take a different plane passing through $P$, then you would obtain a different stress vector using the equation above, thus the stress vector is not unique at a given point.
Getting to your question, the only "true" stress tensor out there is the Cauchy stress tensor we used above, and which relates the unit normal to the plane in the deformed configuration $\vec n$ to the actual stress felt by the body $\vec t$. The Piola-Kirchhoff tensors are so-called "pseudo stress tensors", as they relate different quantities together.
The first Piola-Kirchhoff stress tensor $\mathbf{T_o}$ is defined by:
$$\vec{t_o} = \mathbf{T_o}\vec{n_o},$$
where $\vec{n_o}$ is the unit normal to the plane in the reference configuration, i.e before we applied the forces to deform the body, and $\vec{t_o}$ is the pseudo stress vector defined as:
$$\vec{t_o} = \frac{dA}{dA_o}\vec{t},$$
where $dA$ is the area of a surface element of the plane in the current configuration and $dA_o$ is the area of the same surface element in the reference configuration. This means that $\vec{t_o}$ would be equal to the real stress had the area of surface element remained constant during the deformation.
The second Piola-Kirchhoff stress tensor $\mathbf{\tilde T}$ is defined by:
$$\vec{\tilde t} = \mathbf{\tilde T} \vec{n_o}$$
where $\vec{\tilde t}$ is the pseudo stress vector defined by:
$$\vec{t_o} = \mathbf F \vec{\tilde t}$$
where $\mathbf F$ is the deformation gradient tensor (which I think is equivalent to your $J_\Phi$).
Thus, to summarize, the Piola-Kirchhoff stress tensors are pseudo stress tensors which relate the unit normal to a surface in the reference configuration $\vec n_o$ to some pseudo stress vector which is generally different from the actual stress. Although these tensors do not give the real stress directly, they are still useful tools in continuum mechanics for computing different other quantities.
Sources: check out the very nice book "Introduction to continuum mechanics" by Michael Lai et al.
